Improved Baseline Control-Point Method (Improved Baseline Control-Point Method)

Author: Tianjiang Shuo
Reference: 钱霙婧(2014)《地月空间拟周期轨道上航天器自主导航与轨道保持研究》
Website: https://cislunarspace.cn

Definition

The Improved Baseline Control-Point Method is an orbit keeping control strategy proposed for Earth-Moon libration point weakly stable quasi-periodic orbits. This method introduces clock information correction and navigation constraints based on traditional baseline control-point method, enabling effective orbit keeping under conditions of initial orbital insertion deviation, navigation errors, and actuator errors.

Traditional baseline control-point method assumes nominal trajectory is precisely known and error-free, suitable for strongly stable orbits. However, for weakly stable quasi-periodic orbits near Earth-Moon libration points, initial insertion deviation leads to rapid orbital divergence, making traditional methods unable to guarantee control effectiveness. The improved baseline control-point method solves this problem through:

Target calculation considering navigation errors
Introduction of clock information correction
Constraints from autonomous navigation system requirements

Orbit Keeping Constraints

Dynamic Environment Constraints

Orbital divergence characteristics: Weakly stable orbits are sensitive to initial values; small deviations cause large orbital deviations
Impulse timing constraints: Impulse application timing must be within controllable orbital range
Fuel constraints: Total impulse budget is limited; pulse magnitude and direction need optimization

Actuator Constraints

Thrust direction constraints: Actual thrust direction constrained by attitude control system
Thrust magnitude constraints: Minimum impulse width and maximum thrust constraints
Execution errors: Deviation between actual $\Delta v$ and nominal values

Convergence arc length constraint: Navigation system needs to provide convergent estimates within half orbital period
Accuracy constraint: Navigation accuracy directly affects control effectiveness
Update frequency constraint: Navigation update frequency must match orbit keeping impulse intervals

Algorithm Principles

Basic Principles of Baseline Control-Point Method

The core idea of baseline control-point method: select several baseline points (Control Points) on the orbit, transforming orbit keeping problem into one of making spacecraft satisfy predetermined states at baseline points.

Let nominal baseline trajectory be $\mathcal{T}^*$ , actual orbit be $\mathcal{T}$ . At the $i$ -th baseline point:

Traditional method:

\min \|\mathbf{x}(t_i) - \mathbf{x}^*(t_i)\|

Improved method:

\min \|\mathbf{x}(t_i) - \mathbf{x}^*(t_i)\| + \|\mathbf{e}_{\text{nav}}\| + \|\mathbf{e}_{\text{act}}\|

Where $\mathbf{e}_{\text{nav}}$ is navigation error estimate, $\mathbf{e}_{\text{act}}$ is actuator error estimate.

Clock Information Correction

A key innovation of the improved method is introducing clock information correction. Since weakly stable orbits are sensitive to time, the orbit keeping controller requires precise time synchronization:

Clock bias estimation: Use navigation filter to estimate clock bias $\delta t$
Time correction calculation: Calculate impact of clock bias on orbit
Corrected impulse calculation: Compensate for clock bias in impulse calculation

Corrected impulse calculation formula:

\Delta \mathbf{v}_{\text{corr}} = \Delta \mathbf{v}_{\text{nom}} + \frac{\partial \mathbf{v}}{\partial t}\bigg|_{t_i} \cdot \delta t

The improved method requires collaborative design with autonomous navigation system:

Target point selection: Choose locations with good navigation observability as baseline points
Impulse timing: Ensure impulse applied after navigation update
Accuracy requirements: Set control accuracy thresholds based on navigation accuracy

Algorithm Flow

Initialization

Select $N$ baseline points on nominal trajectory
Determine target state $\mathbf{x}^*_i$ at each baseline point
Set control accuracy threshold $\varepsilon$

Orbit Propagation

From current state, integrate along orbit to next baseline point
Calculate deviation $\Delta \mathbf{x}_i$ between actual and target states
Estimate navigation and actuator errors

Impulse Calculation

Use state transition matrix to calculate impulse sensitivity
Consider clock correction
Calculate minimum impulse to return to state

Execution and Update

Execute impulse maneuver
Update orbit state
Return to step 2 for continued propagation

Simulation Verification

钱霙婧 (2014) verified the improved method's effectiveness through closed-loop simulation:

Simulation Scenario

Target orbit: Earth-Moon L2 quasi-periodic Halo orbit
Initial deviation: Position 10 km, velocity 1 m/s
Navigation error: Position 100 m (1σ)
Actuator error: Velocity increment 1% (1σ)

Simulation Results

Metric	Traditional Method	Improved Method
Orbit deviation control	Cannot converge	< 1 km
Impulse consumption	Divergent	~10 m/s/year
Control period	Not applicable	~7 days

Results show the improved baseline control-point method achieves effective orbit keeping under complex error conditions, with control accuracy meeting mission requirements.

Comparison with Other Methods

Method	Applicable Scenario	Advantages	Disadvantages
X-axis velocity constraint pulse	Halo orbit	Simple, intuitive	Only for specific orbits
Floquet mode method	Periodic orbit	Theoretically complete	Computationally complex
Traditional control-point	Strongly stable orbit	Mature, stable	Not suitable for weakly stable orbits
Improved baseline control-point	Weakly stable quasi-periodic orbit	Considers multi-source errors	Higher computation

References

钱霙婧. 地月空间拟周期轨道上航天器自主导航与轨道保持研究[D]. 哈尔滨工业大学, 2014.
Folta D, Quinn D, Quinn T. Stationkeeping of L2 libration point orbits with ESM manifests[C]. AIAA/AAS Astrodynamics Specialist Conference, 2014.